Astronomical Games: September 2000

How to Cook a Star

The reason stars shine

Wilt thou reach stars, because they shine on thee?

—The Two Gentlemen of Verona, Act III, scene i

I REMEMBER, as a child, going
to the Exploratorium in San Francisco.
The Exploratorium is a hands-on science museum for kids and adults,
where you can explore such disparate science topics as the motion
of objects in a gravitational field, the effect of darkness and delay
on three-dimensional vision, how engines work, and so forth. It's
still there, and a great way to spend a day.

One of the exhibits they had was a sort of primitive pinball machine,
with only one bumper. This bumper had an unknown shape—at least, it
was unknown to the user at first—because it was hidden underneath a
circular mask, which covered it completely. However, you could fire
pinballs up at the bumper, which would strike it underneath the mask,
and then bounce back every which way. You could slide the plunger back
and forth along the bottom edge of the frame, and also rotate the bumper,
so that you could strike the bumper from different positions and angles.

The purpose of all this firing and bouncing was to try to determine
what shape the bumper was. You couldn't see the actual bumper, you
couldn't get at it to take it apart; all you could do was fire pinballs
up at it and observe which way they bounced. They had a whole bunch of
these machines; eventually, you could piece together that the shape was
a star, or a hexagon, or a triangle, or whatever it was. Sometimes you
would decide right away that it was a square, and all of a sudden one of
the pinballs, fired just right, would bounce back in a completely
unexpected direction. You would have to play around a little longer
before you realized that the shape was really a cross.

That, to me, is the essence of science. You practically never have
full and complete access to the internal workings of whatever it is
you're trying to figure out. Either it's too small, or too old and
decayed, or too hard to get at; it's always too something to
look at directly and figure out what's going on. You're always limited
in how you can manipulate the darn thing. And yet, using even
your limited toolset, you can arrive at a more complete picture of how
it works.

In astronomy, of course, the problem is that everything
is too far. Only in this century has it become feasible to travel to the
moon and planets, and even then it is only from the moon that we have
actually gotten anything back and had a chance to look at it. And although
it now appears conceivable to one day travel to the stars, it's hardly out
of the question that we still won't be able to go in and take core
samples. How, then, do we figure out what's in the stars, and what
makes them shine?

For generations upon generations, human observers have looked up at
the stars and been unable to see anything more distinct than dots of light.
It might have been possible for eagle-eyed humans to see the discs of
Venus or Jupiter, to be sure, but even there it was known as early as
the ancient Greeks that these were planets and not the fixed stars,
so of course they were different. The stars remained infinitesimal
points of light.

Even after Galileo first brought a telescope to bear on the stars
(and no, he did not invent it), that's all anyone ever saw:
points of light. If anyone did see the stars as finite in extent, it
was soon revealed that their telescope was in error, and not that they
had actually seen the surface of a star. (Galileo's telescopes were
so inferior that he was unable to distinguish Saturn's rings from its
disc, something that practically any dime-store telescope nowadays can
do.) Centuries passed, telescopes improved in both resolution and
light-gathering power—and points of light the stars steadfastly
remained.

This led French philosopher Auguste Comte (1798–1857), in trying to
come up with an example of a sort of knowledge that would forever be
unattainable, to arrive at the composition of the stars. We would
never be able to sample them physically, he argued, and there seemed
no other way to analyze them, since no one could see the surface
of any star, other than our own sun:

To attain a true idea of the nature and composition of this
science [that is, astronomy], it is indispensable…to mark
the boundaries of the positive knowledge that we are able to
gain of the stars….We can never by any means investigate
their chemical composition….The positive knowledge we can
have of the stars is limited solely to their geometrical and
mechanical properties. [Cours de philosophie positive, 1842]

By "mechanical properties," Comte meant the movement of the stars
through space, not their internal workings. As it sometimes goes,
though, it took only a couple of years after Comte died for him to
be proven wrong, and it didn't in fact require travel to the stars.
Since Isaac Newton (1642–1727), it had been known that white light,
such as light from the sun, could be broken up by a prism into its
constituent colors. This spectrum, as it came to be called, appeared
to be smooth—that is, there were no gaps anywhere where colors might
be missing.

By the 19th century, however, the Bavarian optician Joseph von
Fraunhofer (1787–1826) had discovered that the spectrum was not in
fact completely smooth, but instead had little gaps here and there.
These gaps are now called Fraunhofer lines in his honor. Then, in
1859, the German physicist Gustav Kirchhoff (1824–1887) discovered
that when you heated certain minerals, the light emitted by the
resulting flame wasn't a smooth and complete spectrum, as was light
from the sun. Instead, it was a collection of small bands of light,
each in the right position along the spectrum, but in isolation.

Could it be that the gaps and the light bands were related in some
way? When Kirchhoff looked at a pair of gaps in the solar spectrum,
near the orange portion, he noticed that they seemed to be in exactly
the same position where there were two lines emitted by burning
sodium. To test that they were the same, he decided to pass sunlight
through burning sodium vapor, and then subject the combined light to a
prism, expecting to see the emission lines of the sodium vapor fill in
the gaps in the solar spectrum. To Kirchhoff's great surprise, they
did no such thing. If anything, the gaps became even darker and more
distinct than they had been without the sodium vapor. Were the
emission lines of the sodium vapor just off a bit from the gaps in
the solar spectrum, and somehow mysteriously drawing further light
from those gaps?

It took a few experiments more before Kirchhoff discovered what was
going on. The burning sodium vapor was in fact emitting the same
lines as were missing in the solar spectrum. However, when light from
one source (the sun) passes through another light source cooler than
the first (the burning sodium), the second source absorbs precisely
the same lines it would emit by itself. In recognition of Kirchhoff's
investigations into gas emissions and absorptions, this whole business
about cooler gases absorbing and hotter gases emitting is known as
Kirchhoff's law. In this way, he had proven Comte wrong, just two
years after the latter's death.

Soon after, in 1862, the Swedish physicist Anders Ångstrom (1814–1874)
examined the solar spectrum in much finer detail than Kirchhoff had,
and discovered hydrogen in the sun. Ångstrom measured the position of
the hydrogen lines (and a whole host of others) to a precision of one
ten-billionth of a meter. In his honor, that length is now named after
him, and you will still see, occasionally, light wavelengths measured
in Ångstroms. (The appropriate official metric unit is the nanometer,
which is a billionth of a meter.)

He had to measure it so finely, because as it turns out, hydrogen
produces relatively faint absorption lines, even though it constitutes
most of the matter in the sun (and, indeed, throughout the universe).
The sun, we now know, is about 75 percent hydrogen. The remainder is
mostly an element that was first discovered six years later when the
French astronomer Pierre Janssen (1824–1907) detected absorption lines
that didn't appear to correspond to any element then known. The
British astronomer Joseph Lockyer (1836–1920) examined Janssen's data,
agreed that the element indicated was a previously unknown one, and named
the newly discovered substance helium, after the Greek word for "sun."

Hydrogen and helium are the lightest two elements, which explains why
the earth has so little of them. The earth's gravity is simply
insufficiently strong to hold onto hydrogen and helium—it can only
hold onto the heavier gases, such as nitrogen, oxygen, carbon dioxide,
and so forth. Hydrogen is at least reasonably reactive, so that it
forms lots of compounds that the earth can hold onto, such as methane
and ammonia and water. Helium, on the other hand, is an inert gas; it
tends to "ignore" other elements, and the only reason that we have any
amount of it for blowing up zeppelins and party balloons is that it is
a byproduct of the decay of radioactive elements. (There's an interesting
story about how helium was first discovered on the earth, but that's a
question for another time.) When a party balloon
loses its helium, that helium dissipates into the atmosphere, where it
eventually finds its way outward into space and is lost forever. The
sun, on the other hand, is so massive that although it is very hot, so
that the hydrogen and helium atoms are jostling about very rapidly, its
gravity is more than sufficient to keep those atoms from escaping.

So, the sun is made of about three-quarters hydrogen and one-quarter
helium, and a tiny smattering of other elements. If we mix those gases
here on earth, we certainly don't get the sun. We could light it up, and
then the mixture would likely go up in a blaze of fire (provided enough
oxygen were present), but it would last only for a brief time. How then
does the sun stay lit?

People have been wondering about that for a long time. The earthly
activity most like the shining of the sun is clearly the
burning of fire. Here, of course, fire is dynamic, local in effect, and
temporary in duration, whereas the sun seems steady and wide-ranging,
and as far as anyone can tell, it's been shining essentially forever.
Just the same, it was a reasonable explanation on the face of it, so
before we knew what the sun was made of,
the German physician Julius Mayer (1814–1878) made the calculations
and determined that if the sun were essentially a humongous lump of coal,
there was enough fuel there to last the sun about 5,000 years. Other
substances could be substituted in place of coal, but none of them, in
fact no chemical phenomenon at all, could be relied on to produce the
sun's level of energy production for more than several thousand years,
and in any case the soon-to-be-discovered composition of the sun made
most of these infeasible to begin with.

Mayer himself then proposed another explanation. Perhaps, he suggested,
meteoroids and other space debris were attracted continually by the
sun's gravity and fell to the sun, in the process generating heat. It
had been known for some time that a moving object carries kinetic
energy according to the equation

KE = 1/2 mv2

and when an object comes to a stop, that kinetic energy cannot simply
vanish. It has to be transferred elsewhere, and the "lowest
common denominator" form of energy is heat. This explains why a hammer
gets hot after striking a lot of nails—because the hammer's motion is
impeded by the nails (or, if you're unlucky, your thumb), and the kinetic
energy of the hammer's bulk motion is transferred to the kinetic energy
of the molecules in the hammer (and the nails and your thumb, too, for
that matter). Heat, in other words. Something similar might be happening
with the sun, and if so, maybe that was keeping the sun lit.

Mayer's proposal had the advantage of being able to keep the sun in
continual power, since as far as anyone knew, there could be a limitless
supply of space debris. However, the Irish physicist William Thomson
(1824–1907), later Lord Kelvin, discovered another problem with Mayer's
proposal. When space debris collides with the sun, its kinetic energy
is turned into heat and that can be radiated away. However, the amount
of kinetic energy, according to the above formula, depends not only on
the velocity of the debris but on its mass as well, and that mass cannot
be radiated away. It has to stay in the sun's bowels. Thomson
calculated the rate of mass that would have fall into the sun to support
its current power output, and determined that the sun would have to
"gain weight" so quickly that its gravitational force would have increased
and the orbit of the earth would have shrunk measurably in historic times.
Since no such shrinking had ever been detected, Mayer's proposal had to
be rejected too.

Well, if falling objects couldn't keep the sun on, perhaps the sun was
falling in on itself. The German philospher Immanuel Kant (1724–1804)
had earlier proposed the nebular hypothesis of the sun's formation, which
theorized that the sun and the rest of the solar system condensed out of
a great cloud of gas and dust. If Kant was right and the sun had indeed
collapsed from a cloud of gas and dust, maybe that condensation itself
was sufficient to power the sun. That way, the sun could heat up without
gaining mass and changing the earth's orbit in any way. The German
physicist Hermann von Helmholtz (1821–1894) calculated that under
reasonable assumptions, this process was enough to keep the sun radiating
at its current power for several million years. Unfortunately, by this
time, the earth was known (from geological and biological lines of
reasoning) to be much older than this.

The crucial clue came in the form of the famous theory of general
relativity, developed by the Swiss physicist Albert Einstein (1879–1955).
In it, Einstein developed his famous equation,

E = mc2

which tells us that any mass—a bit of starstuff, an automobile, dryer
lint—can be transformed into an astounding amount of energy. The
British astronomer Arthur Eddington (1882–1944) suggested that the sun
had so much hydrogen and helium because it was turning the hydrogen
into helium. Helium was available in small quantities to study in the
laboratory, and it had been discovered that a helium atom weighed almost
but not quite as much as four hydrogen atoms; the precise ratio was
closer to 3.97 to 1. Perhaps, Eddington mused, the missing mass was
being transformed into energy in the form of light and heat.

Eddington quickly convinced himself and others that the amount of energy
liberated by this transformation was enough to power the sun, at least
in principle. The
sticking point was how, exactly, four hydrogen atoms would come together
to form a single helium atom. Atoms are electrically neutral, so they
have no problem at all bumping into each other—all the more so in the
hot interior of the sun, which Eddington calculated to be 40 million
degrees Kelvin. However, in order to fuse four hydrogen atoms into a
helium atom, it is the atomic nuclei that have to come into contact, and
those are positively charged and instantly repel each other. If the
hydrogen atoms are moving fast enough, the repulsion can be overcome and
the atoms will fuse, but there is a catch: in order to fuse more than a
trivial amount of hydrogen, the temperature had to be much higher than
Eddington calculated—more like
tens of billions of degrees. Such a hot sun is
incompatible with the sun's current size; at those temperatures, the sun
should be much more bloated than it is. Eddington was convinced therefore
that hydrogen was somehow fusing instead at the much colder
temperature of 40 million degrees; he wrote in 1927,

We do not argue with the critic who urges that the stars
are not hot enough for this process; we tell him to go and find a hotter
place.

However, physicists could see no way around the electric
repulsion problem.

As it turned out, however, there was a way. Just about the time that
Eddington was fighting electric repulsion, the young Russian physicist
George Gamow (1904–1968) showed that quantum mechanics explained how
atomic nuclei could split apart in nuclear fission, even though the nuclear
force holding the nucleus together was technically too strong to allow
this to happen. The Welsh astronomer Robert Atkinson (1898–1982) and
German physicist Fritz Houtermans (1903–1966) determined that the same
mechanism could permit hydrogen nuclei to come together, even
though the electromagnetic force holding them apart was
technically too strong to allow this to happen, and they wrote this up
in a paper in 1929. At last, the primary energy source of the stars
was understood.

Let us recap, then. Stars derive their energy from fusion. In the
process of fusing into a single atom of helium, four hydrogen atoms lose
some mass; this mass is transformed into energy. The hydrogen atoms do
not all come together at once—even quantum mechanics does not allow this
in the sun. Instead, the sun fuses hydrogen in two main ways:

In one process, a carbon atom "swallows" the
four hydrogen atoms, one by
one, emitting bits of energy along the way and becoming in turn different
forms of nitrogen and oxygen, until it "burps out" the helium at the end
and returns to its original carbon self. This is known as the carbon
cycle.

In the second process, two hydrogens come together to form deuterium,
a heavy form of hydrogen weighing about twice as much; this then
collides with another hydrogen to form helium-3, a lighter form of helium
which weighs—you guessed it—three times as much as ordinary hydrogen;
finally, two helium-3 atoms collide to yield an ordinary helium atom and two
hydrogen atoms. Those two hydrogen atoms are then available to be fused.
This is known as the proton-proton chain.

Which process is more dominant in any given star depends on that star's
temperature. For our own star and cooler stars, the proton-proton chain
is dominant. Stars much hotter than our own produce most of their energy
via the carbon cycle.

The advent of computers made it possible to simulate stars of different
mass and watch their progress in reasonable times (by human standards).
Several billion years of stellar evolution could be compressed at first
into weeks, then days, and then hours. Out of these simulations came the
conclusion that more massive stars burned their hydrogen fuel much
quicker than less massive ones. Even though they had more hydrogen
to begin with, they also burned hotter, and went through their hydrogen
supply that much quicker. The sun has enough hydrogen to last about
11 billion years of uninterrupted fusion. Much larger stars—say, those
of about 50 times the sun's mass—run out of hydrogen in only a few
million years. On the other hand, much smaller stars, perhaps
as small as a tenth of the sun's mass, run out of their hydrogen supply
only after trillions of years. All of the stars this small
are still burning steadily, because there hasn't been enough time since
the universe was born for them to burn out.

But what happens after the hydrogen runs out? What then? Do they just
fizzle out and gradually cool down, as many astronomers suspected? Or
does something more fantastic happen? When the earliest stellar models
were run through the computers, the simulations didn't yield anything
like what we believed to be aging stars. Nor did they yield any kind
of nonsensical result. They refused to yield anything at all; once the
fusible hydrogen was depleted, the computers were unable to proceed
further. There was hydrogen further out in the star, but
only where it was so cool that fusion couldn't take place. It would
take a completely new simulation technique to carry the life story of
stars forward into death.

In my next essay, then,
I'll talk about what happens when stars run out of hydrogen.

In "OK, Two Stars to Steer Her By,"
I wrote on the mathematics of digital setting
circles (DSCs) and GOTO scopes. The two operate on similar principles,
but they're put together in different ways. Chris Peterson
<clp@alumni.caltech.edu>
wrote to explain that while DSC encoders are placed on the mount axes,
GOTO encoders are placed on the drive motors, before the gear reduction,
so that they know their position to a precision of a fraction of an
arcsecond, at least in theory. (Mount mechanics don't allow such
precision in practice.)

Since I wrote "The Bradley
Firstar" on the fictional Firstar 2000, a new
device has come out on the market that I thought I would mention in
connection with the teaching GOTO. Most laser pointers are red, but
it is also possible, of course, to make green laser pointers. It turns
out that the green lasers are scattered much more effectively than the
red ones, meaning that the green beam can be seen all the way up for
around a kilometer, while red beams can only be seen when they strike a
target. This would make an effective pointing device for classroom
purposes, to indicate where the Firstar is pointed. Current prices are
around $200 or $300, although I would expect them to drop significantly
if demand is sufficient.